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I have completed a course on Discrete Mathematics and really enjoyed studying the chapter on relations. In fact I went back and finished what we hadn't covered in class. I did basic stuff like n-ary relations , representations of relations, closure of relations, equivalence of relations and partial ordering.

I'm an undergraduate and was hoping to get recommendations for online sources and books which discuss more advanced topics on relations.

Or are relations studied under some other field like categorical theory? Any how if you can't think of a specific book, what are some topics and theorems, or books for related topics(other functions) you could point me in the direction of? Thanks a lot and appreciate it.

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This mainly addresses your second question; describe some field where relations are important.

For some reason, Algebraic Combinatorics comes to mind (personal bias -- I like algebra). Among the central objects of study are association schemes, which can be viewed alternatively as sets of relations, sets of graphs, or matrix algebras. There are a number of accessible books, including a very readable one by Godsil (title: Algebraic Combinatorics). His personal page has a frog on it, and I think he is a good writer:

http://www.math.uwaterloo.ca/~cgodsil/

If you are secretly a pure combinatorics person, a somewhat lengthy book is Richard Stanley's Enumerative Combinatorics. Here's a link:

http://math.mit.edu/~rstan/ec/ec1.pdf

Once again, while this may not be exactly what you want, this is very available and certainly has a discrete flavour. You may browse. While the word 'relation' seems only to be used once in this volume, there is the obvious similarity to graphs.

Still more fruitful (perhaps you've already done this) might be to google publications by your book's author (assuming s/he is a research mathematician). If you have good reason to believe that you may have similar interests, this could aid you in your exploration.

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